Some Properties Of Composition Operators And Its Generalized Operators

In recent decades,research of composition operators on holomorphic function spaces has been favored by many mathematical workers.Research of composition operators is the combination of holomorphic function theory and operator theory.One hand,we can use the conclusions of holomorphic function theory to discuss the properties of linear operator.On the other hand,we can also use the operator theory as a tool to study the holomorphic function spaces.The main object of this paper is to study properties of composition operators and the generated operators(weighted composition operators and integral operators).This paper involves five chapters.In the first chapter,firstly,we introduce the research background and present situation of several complex variables,the operators involved in this paper and the related properties.Secondly,we present the main results of this paper.In the second chapter,basic notions,the relative holomorphic function spaces and the operators are presented.The third chapter,we study two classical properties that are boundedness and compactness of composition operators on Zygmund-Orlicz spaces of unit disk.Firstly,we introduce Zygmund-Orlicz spaces by Young's function which extended Zygmund spaces.Secondly,we prove that each Zygmund-Orlicz space is isometrically equal to certain ?-Zygmund space for a very special weight ?.At last,we obtain the main results by using the equivalence of norms of two spaces.In the fourth chapter,boundedness and essential norm of integral operators on ?-Bloch of unit ball are studied.We use power of some function to characterize boundedness and essential norm of integral operators which are different from previous results.The last chapter is focus on commutators of weighted composition operators on Hardy spaces of the unit ball.Here,the weighted composition operators are induced by non isomorphic fractional linear mapping with only one boundary fixed point.We give calculation formula of commutators of weighted composition operators,then we discuss whether commutators are compact or not according to parabolic and hyperbolic fractional linear mapping.At last,we obtain the essential normality of weighted composition operators.